The following article was my ﬁrst venture into the mathematics of com­plex chemical systems. Its aim was to determined what can be said about rate constants and the reaction network itself when one has information about composition dynamics near an equilibrium point. (It’s assumed that the ki­netics is mass action and exhibits detailed balance.) Item [1.C2] is a “pure math” article that was a consequence of things in [1.C1].

The papers [1.C3]–[1.C5] below provide the basis for what is now known as the Deﬁciency Zero Theorem. (Although they do not derive from work by me or my students, the articles by Fritz Horn and Roy Jackson are listed here because they provide the foundation for the Deﬁciency Zero Theorem story.)

In a seminal and beautiful paper [1.C3], Horn and Jackson invented the idea of complex balancing at an equilibrium as a generalization of what chemists call detailed balancing. They showed in [1.C3] that if a mass action system admits a complex balanced equilibrium at which all species concen­trations are positive, then the corresponding differential equations admit only behavior of a special dull kind. In [1.C4] Horn went further; in particular, he examined conditions on rate constants in a mass action system (analogous to the Wegscheider conditions for detailed balancing) that ensure the exis­tence of a positive complex balanced equilibrium. In [1.C5] I showed that, when what has come to be called the deﬁciency of the underlying reaction network is zero, then complex balancing must obtain at every equilibrium, independent of what the kinetics might be.

The following paper is out of chronological sequence, but it is close in spirit to the immediately preceding ones. The paper was in response to a question put to me by Roy Jackson. He wanted to know of a systematic way to state conditions on mass action rate constants that are necessary and sufficient for the existence of a positive equilibrium at which detailed balance obtains. It turns out that there is a nice connnection of this with the underlying network’s deﬁciency and its cycle structure.

The next paper is largely a technical one, but it is a workhorse in many of the reaction network theory papers that followed it. Especially impor­tant is the Appendix. Readers familiar with the mathematics of reaction network theory will perhaps know that, when the kinetics is mass action, the species formation rate function is often written in the decomposed form YAkΨ(c) — see for example §4.A in Lectures on Chem­ical Reaction Networks. The Appendix of [1.C7] deals with the connection between reaction network structure and qualitative properties of the kernel (nullspace) of Ak.

Papers [1.C8] and [1.C9] provide the foundations for deﬁciency one theory. In particular, [1.C8] has a proof of the Deﬁciency One Theorem, and [1.C9] gives the mathematics behind the Deﬁciency One Algorithm, which was im­plemented in even the earliest versions of The Chemical Reaction Network Toolbox. The algorithm converts questions about the capacity of deﬁciency one mass action systems to admit multiple steady states (essentially ques­tions about systems of nonlinear polynomial equations) into questions about systems of linear inequalities, for which there is a well-developed theory.

The next article [1.C10] is out of chronological order, but it has resonances with papers dealing with detailed balancing, complex balancing, the Deﬁ­ciency Zero Theorem, and the Deﬁciency One Theorem. The central question in [1.C10] is about the robustness of steady states: How does reaction network structure alone inﬂuence the sensitivity of the steady state concentration of a particular species to a perturbation in the supply of another species?

It turns out that such sensitivities often have bounds that depend in a striking way on the extent to which the various molecular species are con­structed from a large number of distinct building blocks (as are proteins) and, also, the extent to which those building blocks combine gregariously. I have great affection for this paper because it was the beginning of my col­laboration with Guy Shinar. The paper is an elaboration on work done by Guy when he was a student in Uri Alon’s lab at Weizmann Institute. (The relationship between [1.B4] and [1.C10] is discussed in [1.B5].)

In one way or another the articles [1.C12]-[1.C17] provide mathematical foundations for [1.B.2], the more friendly expository PNAS paper that draws connections between the capacity of a (mass action) network to give multi­ple steady states and the nature of the network’s Species-Reaction Graph. I now think that the self-contained articles [1.C16] and [1.C17] substantially transcend [1.C12]-[1.C15] in terms of results, while at the same time proceed­ing in a different, more economical way. Some readers might want to start there. However, [1.C12]-[1.C15] give an alternative route, based largely on Gheorghe Craciun’s remarkable PhD work — a route that remains highly compelling.

Both [1.C12] and [1.C13] presume mass action kinetics. In this context, [1.C12] provides a determinant condition which, for a given reaction network, serves to ensure injectivity. Injectivity is a network property that, among other things, precludes the possibility of multiple positive steady states, regardless of rate constant values. In turn, [1.C13] provides conditions on a network’s Species-Reaction Graph that suﬃce to ensure that the determinant condi­tion of [1.C12] is satisﬁed. From this, injectivity and multiple-steady-state­-preclusion follow.

In very surprising subsequent work, Gheorghe Craciun and Murad Ba­naji subsequently proceeded in a different way to show that Species-Reaction Graph conditions that suffice for injectivity (and, therefore, the preclusion of multiple steady states) in the context of mass action kinetics also suffice for injectivity for the far wider class they called non-autocatalytic (NAC) kinetics.

However, [1.C12], [1.C13] — and the Craciun-Banaji work — have limita­tions: Crucial to the mathematics is the presumption that the reaction network under consideration is fully open. This amounts to saying that every species is “removed” (perhaps through an unspeciﬁed degradation reaction) at a rate proportional to its current concentration. Although this presumption is natural for continuous ﬂow stirred tank reactors (CFSTRs) in a chemical engineering setting, it is less compelling for models of cellular reaction sys­tems, in which only certain species (or none at all) might suffer degradation. Removal of the fully open presumption makes the mathematics substantially more delicate.

The papers [1.C14] and [1.C15] are aimed at establishing conditions under which results in [1.C12] and [1.C13] can, in fact, be invoked in the absence of the “fully open” presumption. The analysis in [1.C.14] shows that results of the kind given in [1.C12] and [1.C13] can be invoked in quite general settings, provided that one restricts attention to steady states that are, in a certain sense, non-degenerate. The argument in [1.C14] for extending “fully open” results to more general settings does not rely on mass action kinetics. The paper [1.C15] does invoke mass action kinetics, but it gives far sharper results, without limiting considerations to non-degenerate steady states. Among other things, it shows that the results in [1.C12] and [1.C13] do apply even in consideration of systems that are not fully open, provided that the reaction network in question falls within the large class of normal networks — in particular, if it is reversible or even weakly reversible. (A weakly reversible network is one in which every reaction arrow is in at least one directed arrow cycle.)

In many ways, the following papers with Guy Shinar and Daniel Knight subsume results of [1.C12]-[1.C15] and go further. As with the work of Craciun and Banaji, these papers embrace very wide and natural classes of kinetics. But here, from the very beginning, systems that are not “fully open” are admitted for study along­side those that are. In [1.C16] we describe and examine properties of a large class of reaction networks that possess a structural property called concor­dance. (Whether or not a particular network of interest is concordant can be determined by The Chemical Reaction Network Toolbox.)

Among other things, we show in [1.C16] that, in at least some respects, concordance is the structural property that confers upon a network the as­surance of certain dull behaviors against every assignment of kinetics within a very large class called weakly monotonic. In particular, a reaction network exhibits injectivity when taken with every weakly monotonic kinetics if and only if it is concordant. On the other hand, reaction network “discordance” provides a source for unstable behavior in the following sense: For any weakly reversible network that is not concordant, there exists a weakly monotonic kinetics such that the resulting kinetics system exhibits a positive unstable equilibrium. (In [1.C16] we also consider variants of the concordance prop­erty in connection with still wider classes of kinetics, in which certain species might inhibit the occurrence of a particular reaction while others might promote it.)

In [1.C17] and [1.C18] we connect the concordance of a reaction network to properties of its Species-Reaction Graph. In particular, we show that Species-Reaction Graph properties which, in [1.C13], sufficed for injectivity in the fully-open mass action context also suffice for network concordance — and therefore for injectivity against all weakly monotonic (and still broader) kinetics. This is true not only in the fully open context but also whenever the network under study is “nondegenerate,” in particular whenever it is weakly reversible.